Here, \(\mathrm {non}(\mathcal {M})\)  is the uniformity of the meager ideal \(\mathcal {M}\) , that is, the smallest cardinality of a nonmeager subset of the reals, or equivalently, any perfect Polish space. Consequently, in any model of set theory in which \(\mathrm {non}(\mathcal {M})\)  is “large”, i.e., at least \(\aleph _2\) , so is \(\mathfrak {a}_{\mathrm {vec},F}\) . Such models include those satisfying Martin's Axiom \(\mathsf {MA}(\kappa )\)  for \(\kappa \ge \aleph _1\) , as well as the random, Hechler, Laver, and Mathias models (cf. [1]}).
